A reduction of the "cycles plus $K_4$'s" problem
Abstract
Let $H$ be a 2-regular graph and let $G$ be obtained from $H$ by gluing in vertex-disjoint copies of $K_4$. The "cycles plus $K_4$'s" problem is to show that $G$ is 4-colourable; this is a special case of the \emph{Strong Colouring Conjecture}. In this paper we reduce the "cycles plus $K_4$'s" problem to a specific 3-colourability problem. In the 3-colourability problem, vertex-disjoint triangles are glued (in a limited way) onto a disjoint union of triangles and paths of length at most 12, and we ask for 3-colourability of the resulting graph.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.17723
- arXiv:
- arXiv:2406.17723
- Bibcode:
- 2024arXiv240617723D
- Keywords:
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- Mathematics - Combinatorics;
- 05C15